The derivative describes how a function changes at one exact input value. It is the slope of the tangent line to the graph at a point, not the slope over a long interval. This matters because many quantities in science are changing continuously, such as position, temperature, voltage, or population.
The derivative turns a curved graph into local information about direction and rate of change.
The definition of the derivative starts with the slope of a secant line through two nearby points on a curve. If the points are A = (x, f(x)) and B = (x + h, f(x + h)), the secant slope is [f(x + h) - f(x)] / h. As h gets closer to 0, point B moves toward point A and the secant line approaches the tangent line.
If this limit exists, it gives f'(x), the instantaneous rate of change of f at x.
Key Facts
- Derivative definition: f'(x) = lim h->0 [f(x + h) - f(x)] / h
- Difference quotient: [f(x + h) - f(x)] / h
- Secant slope between x and x + h: msec = [f(x + h) - f(x)] / [(x + h) - x]
- Tangent slope at x: mtan = f'(x), if the limit exists
- Instantaneous velocity from position s(t): v(t) = s'(t) = lim h->0 [s(t + h) - s(t)] / h
- A function is not differentiable at a point if the limiting slopes from the left and right do not agree or become infinite.
Vocabulary
- Derivative
- The derivative f'(x) is the limit of the difference quotient and gives the instantaneous rate of change of f at x.
- Difference quotient
- The difference quotient [f(x + h) - f(x)] / h is the average rate of change over a small interval of width h.
- Secant line
- A secant line is a line that passes through two points on a curve and has slope equal to the average rate of change between them.
- Tangent line
- A tangent line is the limiting position of secant lines as the second point approaches the first point.
- Instantaneous rate of change
- An instantaneous rate of change describes how fast a quantity is changing at one exact input value.
Common Mistakes to Avoid
- Substituting h = 0 too early is wrong because the difference quotient usually becomes 0/0 before simplification. First simplify the expression, then take the limit as h approaches 0.
- Confusing secant slope with tangent slope is wrong because a secant line uses two distinct points while a tangent slope is the limit as those points merge. The derivative is not just any average slope.
- Canceling terms incorrectly across addition is wrong because cancellation only works for common factors. For example, (x^2 + 2xh + h^2 - x^2) / h must be simplified to (2xh + h^2) / h before canceling h as a factor.
- Assuming every continuous graph has a derivative is wrong because corners, cusps, and vertical tangents can prevent a single tangent slope from existing. Continuity is necessary for differentiability, but it is not enough.
Practice Questions
- 1 Use the definition f'(x) = lim h->0 [f(x + h) - f(x)] / h to find f'(x) for f(x) = x^2 + 3x.
- 2 For s(t) = 4t^2 - 2t, compute the instantaneous velocity at t = 3 using the derivative definition or an equivalent limit calculation.
- 3 A graph has a sharp corner at x = 2 where the left-hand slope approaches -1 and the right-hand slope approaches 4. Explain whether the derivative exists at x = 2 and why.